Template:Sequence of the Day for December 27

Intended for: December 27, 2012

Timetable

• First draft entered by M. F. Hasler on December 26, 2011 ✓
• Draft reviewed by Alonso del Arte on December 27, 2011 ✓
• Draft to be approved by November 27, 2012

Yesterday's SOTD * Tomorrow's SOTD

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A151542: Generalized pentagonal numbers:

12 n + 3 n  (n  −  1) 2 , n   ≥   0

.

{ 0, 12, 27, 45, 66, 90, 117, 147, 180, 216, 255, 297, 342, 390, 441, 495, 552, ... }

Originally, pentagonal numbers are those of the form

3 n  (n ± 1) 2

.

Those with the “+” sign are sometimes referred to as the “second” pentagonal numbers.

The generalized pentagonal numbers

b n + 3 n  (n  −  1) 2

, for

b = 1

through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

One might wonder whether it is a pure coincidence to find many “remarkable” numbers in this sequence: multiples of 11 (as 66, 297, 495, ...) and other numbers with doubled digits (117, 255 = 2 8  −  1 and its reversal 552, 441 and 882, 1116, 1200, 1377, 1566 and its “permutation” 1665, 2205, 2322, 2442, 3225, 3366, ...), the pair 147 and 741; 216 = 6 3, etc...